Question 5 of 7: EMF Induced in a Loop by a Time-Varying Field
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
17-Phys-A3, Electromagnetics — National Exam, December 2019. 3-hour closed-book exam (Casio or Sharp approved calculators only); any FIVE of the seven questions constitute a complete paper and only the first five as they appear in a candidate's answer book are marked, each of equal value, with full justification required for marks. All seven printed questions are solved below as a complete study resource.
Reference texts: Sadiku, Elements of Electromagnetics (7th ed.) — plane waves and boundaries, transmission lines, waveguides, Gauss's and Faraday's laws; Hayt & Buck, Engineering Electromagnetics (9th ed.) — transmission-line transients and bounce diagrams; Pozar, Microwave Engineering (4th ed.) — transmission-line theory, rectangular waveguide cutoff; Balanis, Antenna Theory (4th ed.) — wave polarization and radiation fundamentals.
Question 5: EMF Induced in a Loop by a Time-Varying Field (20 marks)
Given. $\mathbf B(t)=0.2e^{5t}\,\hat{\mathbf a}_z$ T (part a–d), circular loop of radius $\rho=0.4$ m lying in the $xy$-plane, wire resistance $R=10\ \Omega$. Part (e) repeats with $\mathbf B(t)=0.2e^{5t}\,\hat{\mathbf a}_x$ T, same loop.
Loop in the $xy$-plane with $\mathbf B(t)$ along $+\hat z$ and increasing; by Lenz's law the induced current opposes the growing outward flux, so it circulates CLOCKWISE when viewed from $+z$.
Find. $\text{emf}(t)$, $\nabla\times\mathbf E$, $i(t)$, the current direction, and $\text{emf}(t)$ for the alternative field in part (e).
Approach. Apply Faraday's law in integral form ($\text{emf}=-d\Phi/dt$) with $\Phi=\mathbf B\cdot\hat{\mathbf a}_z\,A$ for the loop's flat area normal to $\hat z$; use the differential (point) form $\nabla\times\mathbf E=-\partial\mathbf B/\partial t$ for part (b); Ohm's law for the current; and re-evaluate the flux integral (not just re-apply the formula) when the field direction changes in part (e).
Part (a) — Flux through the loop and EMF. Loop area $A=\pi\rho^2=\pi(0.4)^2=0.5027\ \text{m}^2$; with the loop normal taken as $\hat{\mathbf a}_z$,
$$\Phi(t)=\mathbf B(t)\cdot\hat{\mathbf a}_z\,A = 0.2e^{5t}(0.5027)=0.1005\,e^{5t}\ \text{Wb}.$$
$$\text{emf}(t)=-\frac{d\Phi}{dt}=-(5)(0.1005)e^{5t}=\boxed{-0.503\,e^{5t}\ \text{V}}.$$
Part (b) — Curl of E. By Faraday's law in point form,
$$\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t}=-\frac{\partial}{\partial t}\left(0.2e^{5t}\right)\hat{\mathbf a}_z=\boxed{-1.0\,e^{5t}\,\hat{\mathbf a}_z\ \text{V/m}^2}.$$
Part (c) — Induced current.
$$i(t)=\frac{\text{emf}(t)}{R}=\frac{-0.503\,e^{5t}}{10}=\boxed{-0.0503\,e^{5t}\ \text{A} = -50.3\,e^{5t}\ \text{mA}}.$$
The negative sign (relative to the $+\hat a_z$-normal right-hand reference direction) means the physical current flow is CLOCKWISE when viewed from $+z$, as sketched: it creates an induced $\mathbf B$ pointing in $-\hat z$ inside the loop, opposing the growing $+\hat z$ applied field — consistent with Lenz's law.
Part (d) — Direction sketch. See the figure above: with the source field increasing along $+\hat z$ (out of the page), the induced current circulates clockwise as viewed from the $+z$ axis, which by the right-hand rule produces a magnetic dipole moment along $-\hat z$ that opposes the increasing external flux.
Part (e): field instead along $\hat{\mathbf a}_x$. The loop's normal is still $\hat{\mathbf a}_z$ (it lies in the $xy$-plane), and $\mathbf B(t)=0.2e^{5t}\hat{\mathbf a}_x$ is now entirely IN the plane of the loop, so it has zero component along the loop's normal at every instant:
$$\Phi(t)=\mathbf B(t)\cdot\hat{\mathbf a}_z\,A = 0,\qquad \boxed{\text{emf}(t)=-\frac{d\Phi}{dt}=0\ \text{for all }t.}$$
No flux ever links a planar loop from a field confined to that loop's own plane, however rapidly that field changes.
